Dynamical and descriptive aspects of groups and their actions

The project lies at the interface of logic and analysis, more specifically, descriptive set theory, measured group theory, graph theory, ergodic theory, probability theory, and operator algebras. In recent years, mathematicians in these fields have come to realize that problems concerning algebraic, dynamical, and descriptive structural complexity of countable groups and equivalence relations can be fruitfully studied via descriptive combinatorial and graph theoretic means. The principal investigator and collaborators have employed this combinatorial perspective to create new tools used to answer several open problems in these fields. This research project aims to generate more new general tools and to promote fruitful interactions among these fields. This project involves training of graduate and undergraduate students. The PI proposes to study problems around orbit equivalence relations, measure equivalence and quasi-isometry, amenability and inner amenability, cocycle superrigidity, treeability, weak equivalence, and rigidity and anti-rigidity for group actions by applying the perspective and insights of descriptive set theory and logic. The problems in the proposal are of interest to many fields, including descriptive set theory, measured group theory, geometric group theory, measurable combinatorics, ergodic theory, probability theory, and operator algebras. The PI and collaborators have already seized the first foothold toward a resolution to many of the proposed problems, and this project is likely to advance knowledge in these fields. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.